Question

which segments are parallel?
select each correct answer.
□ \\(\overline{gh}\\)
□ \\(\overline{ef}\\)
□ \\(\overline{ab}\\)
□ \\(\overline{cd}\\)

Explanation:

Step1: Check $\overline{AB}$ and $\overline{CD}$

Angle at $A$ is $34^\circ$ and angle at $D$ is $34^\circ$. Also, we can check the transversal properties. The alternate interior angles or corresponding angles can be analyzed. Since $\angle A = \angle D = 34^\circ$, and considering the lines $AB$ and $CD$ with transversal $AD$, the corresponding angles are equal, so $\overline{AB} \parallel \overline{CD}$.

Step2: Check $\overline{AB}$ and $\overline{EF}$

At point $E$, the angle adjacent to $138^\circ$ is $180 - 138 = 42^\circ$? Wait, no, let's check the angle at $A$ is $34^\circ$. Wait, maybe using the consecutive interior angles. For $\overline{AB}$ and $\overline{EF}$ with transversal $AE$, $\angle A = 34^\circ$ and the angle at $E$: the angle inside the figure. Wait, maybe another approach. The angle at $E$ is $138^\circ$, and the angle at $A$ is $34^\circ$. Let's check the sum of angles on the same side of transversal. $34 + 138 = 172$? No, maybe alternate interior angles. Wait, actually, $\overline{AB}$ and $\overline{EF}$: let's see the angle at $A$ is $34^\circ$, and if we look at the triangle, the angle at $E$: maybe the corresponding angles. Alternatively, since $\overline{AB} \parallel \overline{CD}$ and we can check $\overline{EF}$ and $\overline{CD}$.

Step3: Check $\overline{EF}$ and $\overline{CD}$

Angle at $H$ is $146^\circ$, and the angle at $D$ is $34^\circ$. $146 + 34 = 180^\circ$, so consecutive interior angles are supplementary. So $\overline{EF} \parallel \overline{CD}$ (since consecutive interior angles are supplementary, lines are parallel). And since $\overline{AB} \parallel \overline{CD}$, then $\overline{AB} \parallel \overline{EF}$ as well. Also, $\overline{GH}$: the angle at $H$ is $146^\circ$, and let's check with other lines. But $\overline{GH}$: the angle sum with other lines. Wait, the angle at $H$ is $146^\circ$, and if we check with $\overline{CD}$, the angle at $D$ is $34^\circ$, $146 + 34 = 180$, but $\overline{GH}$: let's see the length. Wait, no, $\overline{GH}$: the angle at $H$ is $146^\circ$, and the angle at $C$: since $\angle D = 34^\circ$, $\angle C$ should be equal to $\angle B$? Wait, maybe $\overline{GH}$ is not parallel. But $\overline{EF}$, $\overline{AB}$, $\overline{CD}$: let's confirm. For $\overline{EF}$ and $\overline{CD}$: the angle at $H$ is $146^\circ$, and the angle at $D$ is $34^\circ$, $146 + 34 = 180^\circ$, so consecutive interior angles are supplementary, so $\overline{EF} \parallel \overline{CD}$. And since $\overline{AB} \parallel \overline{CD}$, then $\overline{AB} \parallel \overline{EF}$. So $\overline{AB} \parallel \overline{EF}$, $\overline{AB} \parallel \overline{CD}$, $\overline{EF} \parallel \overline{CD}$. So the correct segments are $\overline{EF}$, $\overline{AB}$, $\overline{CD}$.

Answer:

$\overline{EF}$, $\overline{AB}$, $\overline{CD}$ (and also $\overline{AB}$ is parallel to $\overline{CD}$, $\overline{AB}$ is parallel to $\overline{EF}$, $\overline{CD}$ is parallel to $\overline{EF}$)